Differential equations: air reistance

[LukeOD]

Hi, I'm half way through a piece of differential equations coursework and I'm having serious trouble with a couple of things.

The brief of the coursework is to investigate the effects of air resistance on falling objects by conducting an experiment to obtain a set of results, then using differential equations to attempt to model the situation. I have been told to first try the relationship 'R is proportional to v', come up with a differential equation, solve it and then use the resulting equation to come up with a set of predictions which I can compare with my experimental results. I've then been told to do the same using the relationship 'R is proportional to v^2'.

The first problem I've got is that I've done all of this but have no idea whether the equations I've ended up with are correct.

The second problem is that the resulting equations [solved for s because my experimental data are times to fall a given distance] contain a constant of proportionality k (from what I've read I think this is something to do with cross-sectional area and air density, but I' not required to know or use this for the coursework). I wanted to rearrange my equations and solve them for k and then substitute my results in and compare the variation in k, but since I can't do this I've tried substituting an experimental value for time and then using trial and error to find the value of k that gives my experimental value for s. I've then used this value of k to make the rest of the predictions. I've got 3 sets of data for 3 different masses [same cross-sectional area] and I've used a different value of k for each set. The problem is that the both models seem to predict the data fairly well, and I expected only the v^2 to do this.

Should I be using the same value of k for all 3 sets? I've tried this and both models disagree with the results quite a lot, and by roughly the same amount.

Can anyone give me some advice and check my derivations please? I've got them typed up in word but can't post them here because the formatting doesn't work and there's about 6 pages. I can email them or attach them to a PM maybe?

Thanks in advance

EDIT:

this is what I ended up with for 'R is proportional to v'

s = mg/k ( m/k e^(-k/m t)-m/k+t)

and this is what I ended up with for 'R is proportional to v^2'
s = m/k∙ln⁡((e^((k√(mg/k))/m t)+e^(-(k√(mg/k))/m t))/2)
EDIT 2:

apologies for the typo in the title!

 

[hunt_mat]

Use F=ma, then the forces are mg-R because the acceleration and air resistance are in opposite directions, so your basic equation is:

$$mg-kv^{2}=m\frac{dv}{dt}=mv\frac{dv}{dx}$$

Solve the ODE to give v as a function of t. For your second problem, you have to draw a graph and calculate slopes I think.

 

[LukeOD]

Thanks for the reply hunt_mat.
I used the first part of the DE you suggested [mdv/dt=...] and then solved it for v. Once I'd done that I then integrated to get a solution in terms of s.

I used the conditions that s=0 and v=0 when t=0 to obtain the attached solution.


Sorry I can't use LaTeX and I'm having trouble attaching my word 2007 docs. In compatibility mode the whole file is too large to attach so I've just attached the particular solution. Does it look OK?

 

[hunt_mat]

Okay, they look like the sort of think you should be getting.

 

[LukeOD]

Ok thanks.

How do you think would be best to go about using that solution to make some predictions?

I've got 3 sets of experimental data, each using a different mass and each consisting of 5 results [drop height and time]. I've tried using the greatest time from each set and then using trial and error to find the value of k that gives me the corresponding height, then used this value of k for the rest of the set.

I did the same thing for the 'R prop. to v' relationship and graphed them all, but all the graphs look very similar. Really stumped because as far as I know the v^2 should give a much better prediction than the v.

Any ideas?

Thanks in advance.

 

[hunt_mat]

You may get 3 different values of k for the three different sets of data. Take one set and work with that. The graph should look like a tanh curve.

 

[LukeOD]

I've been plotting height against time graphs but the predictions give almost completely straight lines for both v and v^2. The experimental results aren't completely straight but don't look like tanh curves either.

 

[hunt_mat]

When you plot the velocity against time, it should look like tanh.

 

[LukeOD]

Oh ok. If I used the solution in terms of s to find a value for k [using my experimental results], would the same value of k work then in the solution for v?

 

[hunt_mat]

For that particular set of results yes, as I said before, you have three different sets of experimental results and I would expect three different values of k for that result.

 

[LukeOD]

Thanks a lot for your help so far hunt_mat. Sorry to keep asking the same thing but I'm still really stuck on the best way to use the solutions to make the predictions and then compare them.

I've got a solution for s and a solution for v for both relationships. The solution in terms of v for the 'v^2' relationship has a second unknown [which I think is probably terminal velocity] which doesn't occur in the 'v' relationship, but since i don't know any other conditions I'm not sure whether I can find it.

I could send you my experimental results and the full derivations if you want to see them or you think it would help.

Any help will be greatly appreciated.

Thanks again